Question

Q⁺ is the set of all positive rational numbers with the binary operation * defined by $a*b=\frac{ab}{2}$ for all a, b ∈ Q⁺. The inverse of an element a ∈ Q⁺ is
(a) a

(b) $\frac{1}{a}$

(c) $\frac{2}{a}$

(d) $\frac{4}{a}$

Answer 1

(d) $\frac{4}{a}$
Let e be the identity element in Q⁺ with respect to * such that

$$\begin{gathered} a*e=a=e*a,\forall a\in Q^{+} \\ a*e=a\text{and}e*a=a,\forall a\in Q^{+} \\ \frac{ae}{2}=a\text{and}\frac{ea}{2}=a,\forall a\in Q^{+} \\ e=2\in Q^{+},\forall a\in Q^{+} \end{gathered}$$
Thus, 2 is the identity element in Q⁺ with respect to *.

$$\begin{gathered} \text{Let}a\in Q^{+}\text{and}b\in Q^{+}\text{be the inverse of}a. \\ \text{Then,} \\ a*b=e=b*a \\ a*b=e\text{and}b*a=e \\ \frac{ab}{2}=2\text{and}\frac{ba}{2}=2 \\ b=\frac{4}{a}\in Q^{+} \\ \text{Thus,}\frac{4}{a}\text{is the inverse of}a\in Q^{+}. \end{gathered}$$